Matrices And Determinants

Step 1: Understanding the Concept
A system of homogeneous linear equations, represented as , has a non-trivial solution (a solution other than ) if and only if the determinant of the coefficient matrix is zero.
Step 2: Key Formula or Approach
The system of equations is:
We need to find the values of for which .
Step 3: Detailed Explanation
The coefficient matrix is: We set the determinant to zero: Let's compute the smaller determinants. Note that since . Substitute : Multiply the entire equation by (which is non-zero):
Using trigonometric identities and : Multiply by 2 to clear the fraction:
Now, consider the expression . The range of an expression of the form is .
Here, and . The range is .
So, the minimum value of is -5 and the maximum value is 5.
Therefore, the range of the expression is:
Minimum value:
Maximum value:
The expression is always between 6 and 16, so it can never be equal to 0.
Step 4: Final Answer
Since can never be zero, the system of equations only has the trivial solution ( ) for any value of . There is no value of for which a non-trivial solution exists.

Step 1: Understanding the Concept:
We use the properties of the rank of matrices in systems of linear equations (Rouche-Capelli Theorem). For a system with being : 1. Unique Solution: (Full rank). 2. Infinite Solutions: .
Step 2: Analyzing the Data:
* System : Has a unique solution. Since is , this implies . Consequently, the augmented matrix (which is ) must have a rank of 3 because it contains a submatrix with rank 3, and the maximum possible rank for a matrix with 3 rows is 3. So, . * System : Has infinite solutions. Since is , this implies . Also, , where .
Step 3: Evaluating Options:
Now we evaluate . This is an augmented matrix of size . Its rank can be at most 3 (limited by the number of rows). So, . Comparison: We established that . We know that . Therefore, is always true (since ). Let's check other options:

• (A) : Not necessarily true. could have rank 2.
• (B) : False. .
• (C) : . Rank of is max 3 (as it has 3 rows or assuming it's formed by columns, wait are , so is , max rank 2). So is False.

Step 3: Final Answer:

Option (D) is the correct statement.

Step 1: Understanding the Concept
This question involves properties of the adjoint of a matrix. The adjoint of a matrix , denoted as , is the transpose of its cofactor matrix. We will use standard properties relating the adjoint, determinant, and inverse of a matrix.

Step 2: Key Formula or Approach
We will use the following properties for a non-singular square matrix of order :
1.
2.
3.

Step 3: Detailed Explanation
The given matrix is , which is a square matrix of order .
We need to find .

First, find using property (3):
Hence, the expression simplifies to:
Now, we need to find .
For a matrix , we have .
So, for our matrix :
Therefore, .

Now, let's verify using property (2): .
Calculate the determinant of :
Since , exists.
Using property (2):
Since we found , we can conclude:
This matches option (C).

Step 4: Final Answer
Using the properties of adjoint matrices for a matrix of order , we have . Therefore, . Also, . Hence, the correct expression is:

Step 1: Understanding the Concept
A system of non-homogeneous linear equations, represented as , is inconsistent (has no solution) if the determinant of the coefficient matrix is zero, and at least one of the determinants is non-zero. The primary condition to check is .
Step 2: Key Formula or Approach
The given system of equations is: We first write the coefficient matrix and set its determinant to zero to find the values of for which the system does not have a unique solution.
Step 3: Detailed Explanation
The coefficient matrix is:
For the system to be inconsistent, we must have .
This is a quadratic equation in . The problem states that the system is inconsistent for two values, and . These must be the roots of this quadratic equation.
(We should verify that for these roots, the system is indeed inconsistent and not consistent with infinite solutions. This requires checking that , or that at least one of is non-zero. For the roots of , this condition holds, leading to inconsistency).
According to Vieta's formulas for a quadratic equation , the sum of the roots is .
For the equation , the sum of the roots is:
Step 4: Final Answer
The sum of the values of for which the system is inconsistent is .